Powerful Union, Upstart Battle Over Shrinking Pie

OAKLAND, Calif.—The Service Employees International Union is locked in battle here with an unusual opponent: another union.

SEIU has enjoyed years of rapid growth even as organized labor has withered in the U.S. Now, it is competing with the National Union of Healthcare Workers to represent 45,000 nursing aides, pharmacy technicians and janitors at health-care giant Kaiser Permanente. The fight is playing out in cafeterias and break...